A) \[\frac{\pi }{3}\]
B) \[\frac{2\pi }{3}\]
C) \[\frac{\pi }{2}\]
D) \[\frac{\pi }{4}\]
Correct Answer: B
Solution :
Let\[\overrightarrow{a},\overrightarrow{b}\]and\[\overrightarrow{c}\]are three vectors Also, \[\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}\] and \[|\overrightarrow{a}|+|\overrightarrow{b}|=|\overrightarrow{c}|=1\] \[\therefore \] \[\overrightarrow{a}.(\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{a}.\overrightarrow{c}\] \[\Rightarrow \] \[\overrightarrow{a}.\overrightarrow{a}+\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}\] \[\Rightarrow \] \[1+1.1.\cos \theta =\overrightarrow{a}.\overrightarrow{c}\] \[\Rightarrow \] \[1+\cos \theta =\overrightarrow{a}.\overrightarrow{c}\] ?(i) \[\overrightarrow{b}.(\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{b}.\overrightarrow{c}\] \[\Rightarrow \] \[\overrightarrow{b}.\overrightarrow{a}+\overrightarrow{b}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}\] \[\Rightarrow \] \[|\overrightarrow{b}||\overrightarrow{a}|\cos \theta +1=\overrightarrow{b}.\overrightarrow{c}\] \[\Rightarrow \] \[1+\cos \theta =\overrightarrow{b}.\overrightarrow{c}\] ?.(ii) On adding Eqs. (i) and (ii), \[1+\cos \theta +\cos \theta +1=\overrightarrow{a}.\overrightarrow{c}+\overrightarrow{b}.\overrightarrow{c}\] \[\Rightarrow \] \[2(1+\cos \theta )=(\overrightarrow{a}+\overrightarrow{b}).\overrightarrow{c}\] \[\Rightarrow \] \[2(1+\cos \theta )=\overrightarrow{c}.\overrightarrow{c}\] \[\Rightarrow \] \[\cos \theta =\frac{1}{2}-1\] \[\Rightarrow \] \[\cos \theta =-\frac{1}{2}=\cos \frac{2\pi }{3}\] \[\Rightarrow \] \[\theta =\frac{2\pi }{3}\]
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